7ᵗʰ Iranian Geometry Olympiad (Intermediate) P4

For discussing Olympiad level Geometry Problems
IftakharTausifFarhan
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Joined:Sat Dec 12, 2020 3:33 pm
7ᵗʰ Iranian Geometry Olympiad (Intermediate) P4

Unread post by IftakharTausifFarhan » Sat Dec 12, 2020 4:16 pm

Triangle $ABC$ is given. An arbitrary circle with center $J$, passing through $B$ and $C$, intersects the sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $X$ be a point such that triangle $FXB$ is similar to triangle $EJC$ (with the same order) and the points $X$ and $C$ lie on the same side of the line $AB$. Similarly, let $Y$ be a point such that triangle $EYC$ is similar to triangle $FJB$ (with the same order) and the points $Y$ and $B$ lie on the same side of the line $AC$. Prove that the line $XY$ passes through the orthocenter of the triangle $ABC$.

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